In Montgomery-Vaughan's Multiplicative Number Theory I: Classical Theory, they prove in Theorem 15.3 that $\pi(x)-\operatorname{li}(x) = \Omega_{\pm}(x^{\Theta-\epsilon})$ for every $\epsilon>0$ and $\Theta$ being the supremum of the real parts of the non-trivial zeroes of the Riemann zeta function. This in particular shows that $\pi(x)$ and $\operatorname{li}(x)$ cross infinitely many times. In Section 15.3, Montgomery-Vaughan (M-V) write "Theorems 15.2 and 15.3, and Corollary 15.4, are due in substance to E. Schmidt (1903).". However, when I search "pi(x)-li(x) crossover infinitely" in Google, literally every single page cites this infinite crossover result as being from Littlewood (1914): enter image description here

Yes, it is true that Littlewood proved a stronger version of this result in 1914 (Thm. 15.11 in M-V, as they acknowledge again in Section 15.3), but most of the papers/websites just say that Littlewood (1914) showed that there are infinitely many crossovers, when M-V suggest the result was known to Schimdt in 1903.

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    $\begingroup$ So, we need to know what Schmidt proved, and what the phrase "in substance" hides. $\endgroup$ Commented Mar 18, 2022 at 11:08
  • $\begingroup$ Erhard Schmidt, "Über die Anzahl der Primzahlen unter gegebener Grenze," Mathematische Annalen, Vol. 57, No. 2, June 1903, pp. 195-204. (scan online) $\endgroup$
    – njuffa
    Commented Mar 20, 2022 at 1:53
  • $\begingroup$ Séance du 22 Juin 1914, "Sur la distribution des nombres premiers", Note de M. J.-E. Littlewood, présentée par M. J. Hadamard. Comptes Rendus Hebdomaires des Séances de l'Académie des Sciences, Vol. 158, January-June 1914, Paris: Gauthier-Villars 1914, pp. 1869-1872 (BNF-Gallica scan) $\endgroup$
    – njuffa
    Commented Mar 20, 2022 at 2:15

1 Answer 1


The paper Oscillation Error Terms: Littlewood's Result writes

E.Schmidt [1903] could do this under the assumption that the Riemann Hypothesis (RH) is false. Littlewood proved in 1914 that this is also the case when RH is true.

So one could say that while Schmidt's result was part of the proof, it's hard to credit him with having proven the result "in substance" since both results were necessary.

  • $\begingroup$ Strange; that article says that Schmidt established in 1903 that $\psi(x)-x$ has infinitely many crossovers, but it took until Littlewood 1914 to establish that for $\pi(x)-\operatorname{li}(x)$. But in Montgomery-Vaughan both results are proved in the span of 2 pages in Theorem 15.2. Maybe the "in substance" refers to the fact that Schmidt's proof for $\psi(x)-x$ actually can be modified slightly to get the result for $\pi(x)-\operatorname{li}(x)$, but it was Littlewood who first proved the result (using his, more complicated method)? $\endgroup$
    – D.R
    Commented Mar 20, 2022 at 18:45
  • $\begingroup$ @D.R I'm afraid that I'm not familiar with either Littlewood's proof or that of Schmidt. However, when you say that Schmidt's result for $\psi(x) - x$ can be modified slightly to get the result for $\pi(x) - \text{li}(x)$, this step is described by the linked paper's author as "the hard part". If no satisfactory answer is posted here, then you might try sending an email to the paper's author. I have tried to locate his email address but he no longer appears to be at UBC and I was unable to find any contact details elsewhere. $\endgroup$
    – nwr
    Commented Mar 20, 2022 at 20:22

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